update in static_2 and basic_2

[helm.git] / matita / matita / contribs / lambdadelta / static_2 / syntax / tdpos.ma

diff --git a/matita/matita/contribs/lambdadelta/static_2/syntax/tdpos.ma b/matita/matita/contribs/lambdadelta/static_2/syntax/tdpos.ma
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+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "static_2/notation/relations/positive_3.ma".
+include "static_2/syntax/item_sd.ma".
+include "static_2/syntax/term.ma".
+
+(* DEGREE POSITIVITY ON TERMS ***********************************************)
+
+inductive tdpos (h) (o): predicate term ≝
+| tdpos_sort: ∀s,d. deg h o s (↑d) → tdpos h o (⋆s)
+| tdpos_lref: ∀i. tdpos h o (#i)
+| tdpos_gref: ∀l. tdpos h o (§l)
+| tdpos_pair: ∀I,V,T. tdpos h o V → tdpos h o T → tdpos h o (②{I}V.T)
+.
+
+interpretation
+   "context-free degree positivity (term)"
+   'Positive h o T = (tdpos h o T).
+
+(* Basic inversion lemmas ***************************************************)
+
+fact tdpos_inv_sort_aux (h) (o): 
+                        ∀X. 𝐏[h,o]⦃X⦄ → ∀s. X = ⋆s → ∃d. deg h o s (↑d).
+#h #o #H *
+[ #s #d #Hsd #x #H destruct /2 width=2 by ex_intro/
+| #i #x #H destruct
+| #l #x #H destruct
+| #I #V #T #_ #_ #x #H destruct
+]
+qed-.
+
+lemma tdpos_inv_sort (h) (o): ∀s. 𝐏[h,o]⦃⋆s⦄ → ∃d. deg h o s (↑d).
+/2 width=3 by tdpos_inv_sort_aux/ qed-.
+
+fact tdpos_inv_pair_aux (h) (o): ∀X. 𝐏[h,o]⦃X⦄ → ∀I,V,T. X = ②{I}V.T →
+                                 ∧∧ 𝐏[h,o]⦃V⦄ & 𝐏[h,o]⦃T⦄.
+#h #o #H *
+[ #s #d #_ #Z #X1 #X2 #H destruct
+| #i #Z #X1 #X2 #H destruct
+| #l #Z #X1 #X2 #H destruct
+| #I #V #T #HV #HT #Z #X1 #X2 #H destruct /2 width=1 by conj/
+]
+qed-.
+
+lemma tdpos_inv_pair (h) (o): ∀I,V,T. 𝐏[h,o]⦃②{I}V.T⦄ →
+                              ∧∧ 𝐏[h,o]⦃V⦄ & 𝐏[h,o]⦃T⦄.
+/2 width=4 by tdpos_inv_pair_aux/ qed-.
+
+(* Basic properties *********************************************************)
+
+axiom tdpos_total (h): ∀T. ∃o. 𝐏[h,o]⦃T⦄.